1. Field of the Invention
The present invention relates to a polarization evaluation mask for use in evaluation of the polarizing property of exposure equipment, a polarization evaluation method, and a polarization determination device for determining the polarizing property.
2. Description of the Related Art
In recent years, the so-called immersion exposure equipment has been known in the field of semiconductor exposure equipment to improve the resolution for fine patterns. In the immersion exposure equipment, a liquid is filled between the lowermost lens in a projection lens and a semiconductor substrate to realize a projection lens with a NA≧1 (NA means numerical apertures). Because of such the enlarged NA, the reduction in resolution due to polarization of illumination light has been considered as a problem. Accordingly, such a polarized illumination technology has been hastily developed that allows the state of polarization in σ of illumination light to have a special distribution. In addition, a technology has been hastily developed to evaluate the state of polarization of illumination light. A polarization evaluation requires measurements of factors such as an ellipticity of a polarized ellipse, an orientation of the principal axis of a polarized ellipse, the rotational direction of polarization (a right-handed rotation or a left-handed rotation), and a ratio of a polarized component to an unpolarized component in the total intensity (a degree of polarization). Desirably, these factors can be measured simply by the user of exposure equipment unaided without asking the exposure equipment manufacturer for help.
In response to the tendency to shorten the exposure wavelength of the exposure equipment, a calcium fluoride crystal (fluorite) or the like with a high double refraction has been used in the optical system thereof in consideration of the transmissivity. Thus, it is required to measure the magnitude of the double refraction of such the optical system quantitatively. Desirably, this can be also measured simply by the user of exposure equipment unaided without asking the exposure equipment manufacturer for help. Indication of the property of the optical system having a double refraction generally requires the use of a Jones matrix or a Mueller matrix. The Jones matrix is a matrix with 2 rows and 2 columns, which expresses the optical property of an optical element that transmits a completely polarized light expressible with a Jones vector. With amplitude ax, ay and a phase difference δ, the Jones vector expresses an electric field vector (E)=(Ex, Ey) as follows.
                    E        =                              (                                                                                E                    x                                                                                                                    E                    y                                                                        )                    =                                                    ⅇ                                  ⅈ                  ⁢                                                                          ⁢                  ω                  ⁢                                                                          ⁢                  t                                            ⁡                              (                                                                                                                              a                          x                                                ⁢                                                  ⅇ                                                      ⅈφ                            x                                                                                                                                                                                                                            a                          y                                                ⁢                                                  ⅇ                                                      ⅈφ                            y                                                                                                                                              )                                      =                                          ⅇ                                  ⅈ                  ⁡                                      (                                                                  ω                        ⁢                                                                                                  ⁢                        t                                            +                                              φ                        x                                                              )                                                              ⁡                              (                                                                                                    a                        x                                                                                                                                                                          a                          y                                                ⁢                                                  ⅇ                                                      ⅈ                            ⁢                                                                                                                  ⁢                            δ                                                                                                                                              )                                                                        [                  Expression          ⁢                                          ⁢          1                ]            
If the completely polarized light expressible with such the Jones vector is subjected to any conversion when it passes through the optical element, the property of the optical element is expressed as the Jones matrix. For example, a Jones matrix for a polarizer with an orientation angle θ=0° of the transmissive polarization can be expressed as follows.
                              P          ⁡                      (            0            )                          =                  [                                                    1                                            0                                                                    0                                            0                                              ]                                    [                  Expression          ⁢                                          ⁢          2                ]            
A Jones vector can be advantageously computed in a general optical simulator using an electric field vector. Elements of the vector and the conversion matrix are expressed with complex numbers and accordingly can not be met one by one with physical quantities that are directly observable by experiment. In order to combine an actually measured value with a Jones vector, therefore, a proposed method introduces a Pauli matrix that has been used in the quantum mechanics or the like. The introduction seems to complicate the issue in contrast and reduces the significance of purposeful adoption of the Jones vector. The Jones vector supposes only the completely polarized light. Accordingly, it has a fatal defect because it can not respond to a partially polarized light that contains an unpolarized component, which is rather general as the state of polarization in the semiconductor exposure equipment.
Therefore, for expression of the double refractive property of the optical system including a fluorite lens that inevitably causes an unpolarized component, a Mueller matrix is suitable because it can treat the unpolarized component as well. The Mueller matrix is a matrix with 4 rows and 4 columns, which expresses the optical property of an optical element that transmits a partially polarized light expressible with Stokes parameters.
With s0 for the total intensity of light, s1 for the intensity of a 0° linear polarized component (X-axis of the Poincare sphere shown in FIG. 25), s2 for the intensity of a 45° linear polarized component (Y-axis of the Poincare sphere shown in FIG. 25), and s3 for the intensity of a right-handed rotational, circular polarized component (Z-axis (North pole) of the Poincare sphere shown in FIG. 25), the Stokes parameters can be expressed as follows, particularly for the completely polarized light.
                    S        =                              (                                                                                s                    0                                                                                                                    s                    1                                                                                                                    s                    2                                                                                                                    s                    3                                                                        )                    =                      (                                                                                                                                                                                    a                            x                            2                                                    +                                                      a                            y                            2                                                                                                                                                                                                                    a                            x                            2                                                    -                                                      a                            y                            2                                                                                                                                                                                                                2                    ⁢                                                                                  ⁢                                          a                      x                                        ⁢                                          a                      y                                        ⁢                    cos                    ⁢                                                                                  ⁢                    δ                                                                                                                    2                    ⁢                                                                                  ⁢                                          a                      x                                        ⁢                                          a                      y                                        ⁢                    sin                    ⁢                                                                                  ⁢                    δ                                                                        )                                              [                  Expression          ⁢                                          ⁢          3                ]            
On the Poincare sphere, the longitude corresponds to double the orientation angle θ of polarization, and the latitude to double the ellipticity angle ε (see FIG. 25). In a word, any state of polarization can be expressed by one point in a surface of the Poincare sphere.
In the Stokes parameters, (s02−(s12+s22+s32))1/2 expresses an unpolarized component. In a word, the Stokes parameters can also express the partially polarized light that also contains an unpolarized component. In other words, the magnitude of the Poincare sphere expresses the degree of the partially polarized light. In the completely polarized light, S0 matches the radius of the Poincare sphere, and the larger the unpolarized component, the smaller the Poincare sphere becomes. A degree of polarization V, which is an index indicative of the extent of light polarized, can be defined as below.
                    V        =                                                            s                1                2                            +                              s                2                2                            +                              s                3                2                                                          s            0                                              [                  Expression          ⁢                                          ⁢          4                ]            
A conversion of a certain state of polarization expressed with the Stokes parameters to another state of polarization can be expressed using a Mueller matrix with 4 rows and 4 columns. For example, when a light in the state of polarization (S) is converted into another state of polarization (S′) while it passes through an optical element as shown in FIG. 26A, a Mueller matrix (M) for this optical element is used to establish the following expression.S′=M·S  [Expression 5]If plural optical systems are aligned serially as shown in FIG. 26B, the following expression is established.S′=M2M1S  [Expression 6]Accordingly, once the Mueller matrix for the optical element is grasped, the state of polarization after transmission can be precisely predicted through computation from the state of polarization before incidence even though any number of optical elements are stacked. For example, a Mueller matrix for a polarizer with an orientation angle θ of the transmissive polarization and a polarization intensity ratio (1:χ) can be expressed by the following expression.
                              P          θ                =                              1            2                    ⁢                      (                                                                                1                    +                                          χ                      2                                                                                                                                  (                                              1                        -                                                  χ                          2                                                                    )                                        ⁢                    cos                    ⁢                                                                                  ⁢                    2                    ⁢                    θ                                                                                                              (                                              1                        -                                                  χ                          2                                                                    )                                        ⁢                    sin                    ⁢                                                                                  ⁢                    2                    ⁢                    θ                                                                    0                                                                                                                        (                                              1                        -                                                  χ                          2                                                                    )                                        ⁢                    cos                    ⁢                                                                                  ⁢                    2                    ⁢                    θ                                                                                                                                      (                                                  1                          +                                                      χ                            2                                                                          )                                            ⁢                                              cos                        2                                            ⁢                                                                                          ⁢                      2                      ⁢                      θ                                        +                                          2                      ⁢                                                                                          ⁢                      χ                      ⁢                                                                                          ⁢                                              sin                        2                                            ⁢                      2                      ⁢                      θ                                                                                                                                                          (                                                  1                          -                          χ                                                )                                            2                                        ⁢                    sin                    ⁢                                                                                  ⁢                    2                    ⁢                    θcos                    ⁢                                                                                  ⁢                    2                    ⁢                    θ                                                                    0                                                                                                                        (                                              1                        -                                                  χ                          2                                                                    )                                        ⁢                    sin                    ⁢                                                                                  ⁢                    2                    ⁢                    θ                                                                                                                                      (                                                  1                          -                          χ                                                )                                            2                                        ⁢                    sin                    ⁢                                                                                  ⁢                    2                    ⁢                    θ                    ⁢                                                                                  ⁢                    cos                    ⁢                                                                                  ⁢                    2                    ⁢                    θ                                                                                                                                      (                                                  1                          +                                                      χ                            2                                                                          )                                            ⁢                                              sin                        2                                            ⁢                      2                      ⁢                      θ                                        +                                          2                      ⁢                      χ                      ⁢                                                                                          ⁢                                              cos                        2                                            ⁢                      2                      ⁢                      θ                                                                                        0                                                                              0                                                  0                                                  0                                                                      2                    ⁢                    χ                                                                        )                                              [                  Expression          ⁢                                          ⁢          7                ]            
A Mueller matrix for a quarter-wave plate with an orientation angle θ of the fast axis and a retardation Δ=π/2+δ(δ<<1) can be expressed by the following expression.
                                                                        Q                θ                            =                            ⁢                              (                                                                            1                                                              0                                                              0                                                              0                                                                                                  0                                                                                      1                        -                                                  (                                                      1                            -                                                                                                                                                              (                                                  1                          -                                                                                                                                                              -                          sin                                                ⁢                                                                                                  ⁢                        Δsin                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                                                                                                                                                                                                  cos                            ⁢                                                                                                                  ⁢                            Δ                                                    )                                                ⁢                                                  sin                          2                                                ⁢                        2                        ⁢                        θ                                                                                                                                                                  cos                            ⁢                                                                                                                  ⁢                            Δ                                                    )                                                ⁢                        sin                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    cos                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                                                                                                                    0                                                                                      (                                                  1                          -                                                                                                                                    1                        -                                                  (                                                      1                            -                                                                                                                                                              sin                        ⁢                                                                                                  ⁢                        Δcos                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                                                                                                                                                                                                  cos                            ⁢                                                                                                                  ⁢                            Δ                                                    )                                                ⁢                        sin                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                                                  cos                            ⁢                                                                                                                  ⁢                            Δ                                                    )                                                ⁢                                                  cos                          2                                                ⁢                        2                        ⁢                        θ                                                                                                                                                                                                                                                                                                                                                                        cos                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                                                                                                                                                                                                                                0                                                                                      sin                        ⁢                                                                                                  ⁢                        Δ                        ⁢                                                                                                  ⁢                        sin                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                      -                          sin                                                ⁢                                                                                                  ⁢                        Δ                        ⁢                                                                                                  ⁢                        cos                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                            cos                        ⁢                                                                                                  ⁢                        Δ                                                                                            )                                                                                        ≈                            ⁢                              (                                                                            1                                                              0                                                              0                                                              0                                                                                                  0                                                                                      1                        -                                                                              (                                                          1                              +                              δ                                                        )                                                    ⁢                                                                                                          ⁢                                                      sin                            2                                                    ⁢                                                                                                          ⁢                          2                          ⁢                          θ                                                                                                                                                              (                                                      1                            +                            δ                                                    )                                                ⁢                        sin                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                        ⁢                                                                                                  ⁢                        cos                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                      -                          sin                                                ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                        0                                                                                                                (                                                      1                            +                            δ                                                    )                                                ⁢                        sin                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                        ⁢                                                                                                  ⁢                        cos                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                            1                        -                                                                              (                                                          1                              +                              δ                                                        )                                                    ⁢                                                      cos                            2                                                    ⁢                          2                          ⁢                          θ                                                                                                                                    cos                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                        0                                                                                      sin                        ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                                                      -                          cos                                                ⁢                                                                                                  ⁢                        2                        ⁢                        θ                                                                                                            -                        δ                                                                                            )                                                                        [                  Expression          ⁢                                          ⁢          8                ]            
Preferably, a Mueller matrix for an optical system can also be measured simply by the user of exposure equipment unaided without asking the exposure equipment manufacturer for help. The inventors of the present application have already proposed such the exposure equipment, for example, in Patent document 1 (JP 2005/116732 A). With this equipment, the user of semiconductor exposure equipment can measure a Mueller matrix for an optical system contained inside without asking the manufacturer for help.
The equipment of Patent document 1, however, is not of the type that can measure the state of polarization of illumination light itself. Therefore, the measurement of the state of polarization of illumination light must be executed asking the exposure equipment manufacturer for help and thus there is no way to evaluate it by the user's own efforts. It is also difficult by the user's own efforts to evaluate the measurement of the state of polarization of illumination light that is transmitted through an optical system contained inside and illuminated onto a transfer substrate.